The school nurse thinks the average height of 7th graders has increased. The average height of a 7th grader five years ago was 145 cm. She takes a random sample of 101 students and finds that the average height of her sample is 147 cm with a standard deviation of 20 cm. Determine the critical value for testing whether the 7th graders now are taller than they were before? Use a = 0.05.
-1.984
1.984
1.660
-1.660
2.132
The Correct Answer and Explanation is :
To determine the critical value for testing whether the average height of 7th graders has increased, we need to conduct a one-tailed hypothesis test using a t-distribution because the population standard deviation is unknown, and we have a sample size of 101.
Step 1: Define the Hypotheses
- Null Hypothesis ((H_0)): The average height of 7th graders has not increased. That is:
[
H_0: \mu = 145 \text{ cm}
] - Alternative Hypothesis ((H_a)): The average height of 7th graders has increased.
[
H_a: \mu > 145 \text{ cm}
]
Since we are testing for an increase, this is a right-tailed test.
Step 2: Identify Significance Level and Degrees of Freedom
The significance level is given as ( \alpha = 0.05 ).
Degrees of freedom (( df )) for a t-test is given by:
[
df = n – 1 = 101 – 1 = 100
]
Step 3: Find the Critical Value
Since this is a right-tailed test, we need to find the critical value of ( t ) for ( df = 100 ) at ( \alpha = 0.05 ).
From a t-table, the critical t-value for ( df = 100 ) at ( \alpha = 0.05 ) (right-tailed test) is:
[
t_{0.05, 100} = 1.660
]
Step 4: Conclusion
The correct answer is 1.660.
Explanation:
- The t-distribution is used because the population standard deviation (( \sigma )) is unknown.
- The right-tailed test is chosen since we are testing for an increase in height.
- Using the t-table, we find that the critical value at 100 degrees of freedom and ( \alpha = 0.05 ) is 1.660.
- If the calculated test statistic exceeds 1.660, we reject the null hypothesis and conclude that the average height has significantly increased.
- If the test statistic is less than 1.660, we fail to reject the null hypothesis, meaning we do not have enough evidence to support that 7th graders are taller today.
Thus, the correct answer is 1.660.